Why were European monarchs willing to commit their own troops and resources to restoring France's

Louis XVI to power?

Answer: they were afraid of export of revolutionary ideas to their countries. Such an export really took place. At the same time also counter-revolutionary theories and ideas were spreading across Europe. Such an export and its reception is one of the main causes of the birth European constitutionalism. So European monarchs were afraid of the very existence of their dynasties and property and the very basis of political order. From their perspective restoration of Bourbon monarchy was absolutely necessary. By the way the same was the program of Vienna conference in 1814 and 1815.

Explanation: French revolution stimulated significantly development of press and journalism, i. e. political revolution stimulated also social and informational revolution. New ideas were spreading all over France quite quickly and abroad as well.